{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:erdos-654",
  "title": "Erdős Problem #654",
  "canonical_statement": "If $n$ planar points have no four concyclic, must some point determine $(1 - o(1))n$ distinct distances? Failing that, can one always force more than $(1/3 + c)n$?",
  "plain_summary": "VibeMathed reports this item as open. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "open",
  "solution_events": [
    {
      "id": "vibemathed:erdos-654:event",
      "problem_id": "vibemathed:erdos-654",
      "type": "disproved",
      "status": "open",
      "occurred_at": "2026-02-01T00:00:00.000Z",
      "title": "Aletheia project report (Feng et al.)",
      "summary": "the strongest form is disproved via configurations where every point sees at most about 3n/4 distinct distances; the weaker improvement remains open",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:erdos-654:attempt"
      ],
      "method_family_ids": [
        "vibemathed:erdos-654:method"
      ],
      "source_assertion_ids": [
        "vibemathed:erdos-654:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:erdos-654:attempt",
      "problem_id": "vibemathed:erdos-654",
      "model": "Aletheia (Gemini Deep Think)",
      "provider": "Google DeepMind",
      "attempted_at": "2026-02-01T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": null,
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "Aletheia project report (Feng et al.)",
          "url": "https://math.berkeley.edu/~fengt/Aletheia.pdf",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Erdős Problem #654",
          "url": "https://vibemathed.com/problem/erdos-654",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:erdos-654:verification",
      "problem_id": "vibemathed:erdos-654",
      "solution_event_id": "vibemathed:erdos-654:event",
      "level": "independent_expert_verified",
      "mathematical_correctness": "supported",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Expert-reviewed within the Aletheia project, with public report and transcripts; no journal review.",
      "sources": [
        {
          "label": "VibeMathed: Erdős Problem #654",
          "url": "https://vibemathed.com/problem/erdos-654",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "Aletheia project report (Feng et al.)",
          "url": "https://math.berkeley.edu/~fengt/Aletheia.pdf",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "Aletheia project report (Feng et al.)",
      "url": "https://math.berkeley.edu/~fengt/Aletheia.pdf",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Erdős Problem #654",
      "url": "https://vibemathed.com/problem/erdos-654",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:erdos-654:signal:watch_only",
      "problem_id": "vibemathed:erdos-654",
      "kind": "watch_only",
      "reason": "Current public evidence does not meet the default replay threshold.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "uncertainties": [
    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented.",
    "VibeMath has not independently audited the mathematical statement, proof, or novelty claim."
  ],
  "generated_at": "2026-09-13T16:28:40.178Z"
}